Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13172 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(19,25)(20,29)(21,23)(22,35)(26,39)(27,43)(28,34)(30,36)(31,45)(32,44)(33,46)(37,41)(42,49)(47,48);(14,16,17,15)(18,19)(20,26)(21,34)(22,27)(23,25)(24,28)(29,41)(30,40)(31,47)(32,37)(33,42)(35,38)(36,43)(39,45)(44,49)(46,48);(18,20,30,44,23,32,22,29,24,33,35,45,21,31,36,46)(19,26,40,49,25,37,27,41,28,42,38,39,34,47,43,48)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6160);